Validation

The algorithms in ZENO have been extensively tested previously, e.g., Refs. [8][9]. However, in order to validate the implementation of the current version of the code, as well as provide examples for users, five different systems were considered: (1) two touching spheres of radius 1, (2) two touching spheres: one of radius 1 and one of \(1/4\), (3) a cube, (4) a cuboid of 1 x 2 x 3, (5) a “polymer” composed of 20 beads, (6) the protein lysozyme, (7) a torus with a minor radius of 1 and a major radius of 4 that is represented by 416 spheres (8) two touching spheres: one of radius 1 and one of 4 that has been voxelized such that 32 voxels have a length of 1. For lysozyme, the pdb file 1LYD (see Refs. [1][2]) was used, and a 5 \(\unicode{xC5}\) sphere was placed at the center of each alpha carbon of each amino acid. This procedure was previously used in Ref. [7]. All .bod files used in testing can be found in src/cpp/SelfTests.

To validate the accuracy of ZENO, we use analytic values where possible to establish a true value, or ground truth. For systems composed of two spheres or a torus, the analytic values are taken from Ref. [9]. For the cube and cuboid, only analytic values are known for the volume and eigenvalues of the gyration tensor. In all other cases, ground truths do not exist so the output associated with 1e11 walks and 1e11 interior samples was used as a surrogate. In thses cases, the ground truth is taken as the mean from three runs, and the accuracy is determined using the expanded uncertainty (see below) from those three runs. Note that the \(t\) statistic is modified (\(t=4.302653\)). In cases where the representation will generate error, i.e. a torus represented by a collection of spheres and two spheres that have been voxelized, the results are compared to both analytic values where the uncertainity is due to both longer runs of ZENO where the uncertainity is due to the algorithm and the representation and the algorithm.

For each property, the following quantities were calculated: the mean (\(\bar{y}=\sum_{i} y_{i}/N\)), difference between ground truth and mean (\(\Delta=|\mu-\bar{y}|\)), relative difference (\({(\Delta / \mu) \cdot 100\,\%}\)), standard deviation (\(s=\sum_{i} (y_{i}-\bar{y})^{2}/(N-1)\)), standard uncertainty (\(s/\sqrt{N}\)), expanded uncertainty (\(st/\sqrt{N}\)), and relative expanded uncertainty (\(st/(\mu\sqrt{N}) \cdot 100\,\%\)). For the equations, the ground truth is represented by \(\mu\), the number of samples is \(N\), and the value of observation \(i\) of a property is \(y_{i}\). The expanded uncertainty was computed using a 95 % confidence interval (\(t=2.009575\)). Each of the properties was found to have a normal distribution. As a result, the mean and standard deviation should define the distribution.

Results from the aforementioned tests can be found in the following tables. Each test was run a total of 50 different times for 1e6 walks and 1e6 interior samples, as well as 1e7 walks and 1e7 interior samples. Note that these two quantities should be chosen carefully as they affect the uncertainty of the calculation; choosing larger values reduces the expanded uncertainty.

Two touching spheres of radius 1

Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 1.38629 L 1.38628 7.76e-06 0.001% 8.92e-04 1.26e-04 2.53e-04 0.018%
Eigenvalue of \(\mathbf{\alpha}\) 22.658 L \(^{3}\) 22.597 6.14e-02 0.271% 6.40e-02 9.06e-03 1.82e-02 0.080%
Eigenvalue of \(\mathbf{\alpha}\) 22.658 L \(^{3}\) 22.712 5.37e-02 0.237% 6.85e-02 9.69e-03 1.95e-02 0.086%
Eigenvalue of \(\mathbf{\alpha}\) 60.421 L \(^{3}\) 60.432 1.07e-02 0.018% 1.42e-01 2.01e-02 4.05e-02 0.067%
\(\langle\mathbf{\alpha}\rangle\) 35.246 L \(^{3}\) 35.247 6.84e-04 0.002% 6.07e-02 8.58e-03 1.72e-02 0.049%
\(R_{h}\) 1.38629 L 1.38628 7.76e-06 0.001% 8.92e-04 1.26e-04 2.53e-04 0.018%
\(V\) 8.377580 L \(^{3}\) 8.378480 9.00e-04 0.011% 1.07e-02 1.52e-03 3.05e-03 0.036%
\(C_{0}\) 1.25992 L 1.25997 4.59e-05 0.004% 5.38e-04 7.61e-05 1.53e-04 0.012%
Eigenvalue of \(\mathbf{S}\) 0.200000 L \(^{2}\) 0.199257 7.43e-04 0.371% 1.69e-04 2.39e-05 4.81e-05 0.024%
Eigenvalue of \(\mathbf{S}\) 0.200000 L \(^{2}\) 0.200345 3.45e-04 0.173% 2.98e-04 4.22e-05 8.48e-05 0.042%
Eigenvalue of \(\mathbf{S}\) 1.200000 L \(^{2}\) 1.199128 8.72e-04 0.073% 1.49e-03 2.11e-04 4.24e-04 0.035%
[\(\sigma\)] 4.2072 1 4.2068 3.83e-04 0.009% 8.55e-03 1.21e-03 2.43e-03 0.058%
[\(\eta\)] 3.4499 1 3.4469 2.95e-03 0.086% 6.95e-03 9.83e-04 1.98e-03 0.057%

Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 1.38629 L 1.38635 6.45e-05 0.005% 2.96e-04 4.18e-05 8.40e-05 0.006%
Eigenvalue of \(\mathbf{\alpha}\) 22.658 L \(^{3}\) 22.640 1.79e-02 0.079% 1.87e-02 2.65e-03 5.32e-03 0.023%
Eigenvalue of \(\mathbf{\alpha}\) 22.658 L \(^{3}\) 22.679 2.05e-02 0.091% 1.74e-02 2.46e-03 4.94e-03 0.022%
Eigenvalue of \(\mathbf{\alpha}\) 60.421 L \(^{3}\) 60.433 1.23e-02 0.020% 5.04e-02 7.12e-03 1.43e-02 0.024%
\(\langle\mathbf{\alpha}\rangle\) 35.246 L \(^{3}\) 35.251 4.65e-03 0.013% 2.04e-02 2.89e-03 5.80e-03 0.016%
\(R_{h}\) 1.38629 L 1.38635 6.45e-05 0.005% 2.96e-04 4.18e-05 8.40e-05 0.006%
\(V\) 8.377580 L \(^{3}\) 8.377027 5.53e-04 0.007% 4.28e-03 6.05e-04 1.22e-03 0.015%
\(C_{0}\) 1.25992 L 1.25989 2.67e-05 0.002% 2.15e-04 3.04e-05 6.10e-05 0.005%
Eigenvalue of \(\mathbf{S}\) 0.200000 L \(^{2}\) 0.199844 1.56e-04 0.078% 1.04e-04 1.47e-05 2.96e-05 0.015%
Eigenvalue of \(\mathbf{S}\) 0.200000 L \(^{2}\) 0.200107 1.07e-04 0.054% 9.35e-05 1.32e-05 2.66e-05 0.013%
Eigenvalue of \(\mathbf{S}\) 1.200000 L \(^{2}\) 1.199926 7.44e-05 0.006% 4.92e-04 6.96e-05 1.40e-04 0.012%
[\(\sigma\)] 4.2072 1 4.2080 8.16e-04 0.019% 3.36e-03 4.75e-04 9.55e-04 0.023%
[\(\eta\)] 3.4499 1 3.4479 1.97e-03 0.057% 2.74e-03 3.87e-04 7.79e-04 0.023%

Two touching spheres one of radius 1 and one of radius 1/4

Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 1.01992 L 1.01990 1.76e-05 0.002% 4.89e-04 6.92e-05 1.39e-04 0.014%
Eigenvalue of \(\mathbf{\alpha}\) 12.621 L \(^{3}\) 12.594 2.67e-02 0.211% 2.47e-02 3.50e-03 7.03e-03 0.056%
Eigenvalue of \(\mathbf{\alpha}\) 12.621 L \(^{3}\) 12.650 2.85e-02 0.226% 2.50e-02 3.54e-03 7.10e-03 0.056%
Eigenvalue of \(\mathbf{\alpha}\) 14.885 L \(^{3}\) 14.879 5.65e-03 0.038% 3.12e-02 4.41e-03 8.86e-03 0.060%
\(\langle\mathbf{\alpha}\rangle\) 13.376 L \(^{3}\) 13.374 1.60e-03 0.012% 1.58e-02 2.23e-03 4.48e-03 0.034%
\(R_{h}\) 1.01992 L 1.01990 1.76e-05 0.002% 4.89e-04 6.92e-05 1.39e-04 0.014%
\(V\) 4.254240 L \(^{3}\) 4.255200 9.60e-04 0.023% 2.89e-03 4.09e-04 8.22e-04 0.019%
\(C_{0}\) 1.00518 L 1.00526 7.69e-05 0.008% 2.28e-04 3.22e-05 6.47e-05 0.006%
Eigenvalue of \(\mathbf{S}\) 0.197115 L \(^{2}\) 0.196705 4.10e-04 0.208% 1.75e-04 2.48e-05 4.97e-05 0.025%
Eigenvalue of \(\mathbf{S}\) 0.197115 L \(^{2}\) 0.197539 4.24e-04 0.215% 3.82e-04 5.41e-05 1.09e-04 0.055%
Eigenvalue of \(\mathbf{S}\) 0.220784 L \(^{2}\) 0.221115 3.31e-04 0.150% 1.82e-04 2.57e-05 5.16e-05 0.023%
[\(\sigma\)] 3.1442 1 3.1431 1.13e-03 0.036% 4.16e-03 5.88e-04 1.18e-03 0.038%

Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 1.01992 L 1.01995 2.73e-05 0.003% 1.41e-04 1.99e-05 4.00e-05 0.004%
Eigenvalue of \(\mathbf{\alpha}\) 12.621 L \(^{3}\) 12.612 8.61e-03 0.068% 6.87e-03 9.72e-04 1.95e-03 0.015%
Eigenvalue of \(\mathbf{\alpha}\) 12.621 L \(^{3}\) 12.632 1.08e-02 0.086% 7.03e-03 9.94e-04 2.00e-03 0.016%
Eigenvalue of \(\mathbf{\alpha}\) 14.885 L \(^{3}\) 14.885 4.82e-04 0.003% 1.06e-02 1.49e-03 3.00e-03 0.020%
\(\langle\mathbf{\alpha}\rangle\) 13.376 L \(^{3}\) 13.377 5.72e-04 0.004% 4.20e-03 5.94e-04 1.19e-03 0.009%
\(R_{h}\) 1.01992 L 1.01995 2.73e-05 0.003% 1.41e-04 1.99e-05 4.00e-05 0.004%
\(V\) 4.254240 L \(^{3}\) 4.254404 1.64e-04 0.004% 1.32e-03 1.87e-04 3.76e-04 0.009%
\(C_{0}\) 1.00518 L 1.00519 1.43e-05 0.001% 1.04e-04 1.47e-05 2.96e-05 0.003%
Eigenvalue of \(\mathbf{S}\) 0.197115 L \(^{2}\) 0.197031 8.41e-05 0.043% 9.48e-05 1.34e-05 2.69e-05 0.014%
Eigenvalue of \(\mathbf{S}\) 0.197115 L \(^{2}\) 0.197225 1.10e-04 0.056% 8.26e-05 1.17e-05 2.35e-05 0.012%
Eigenvalue of \(\mathbf{S}\) 0.220784 L \(^{2}\) 0.220819 3.48e-05 0.016% 8.87e-05 1.25e-05 2.52e-05 0.011%
[\(\sigma\)] 3.1442 1 3.1442 2.91e-05 0.001% 1.45e-03 2.05e-04 4.13e-04 0.013%

Cube

Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 0.660679 L 0.660727 4.82e-05 0.007% 3.74e-04 5.29e-05 1.06e-04 0.016%
Eigenvalue of \(\mathbf{\alpha}\) 3.6443 L \(^{3}\) 3.6319 1.24e-02 0.339% 7.24e-03 1.02e-03 2.06e-03 0.056%
Eigenvalue of \(\mathbf{\alpha}\) 3.6443 L \(^{3}\) 3.6468 2.49e-03 0.068% 7.01e-03 9.92e-04 1.99e-03 0.055%
Eigenvalue of \(\mathbf{\alpha}\) 3.6444 L \(^{3}\) 3.6601 1.57e-02 0.432% 6.37e-03 9.01e-04 1.81e-03 0.050%
\(\langle\mathbf{\alpha}\rangle\) 3.64431 L \(^{3}\) 3.64629 1.98e-03 0.054% 5.28e-03 7.46e-04 1.50e-03 0.041%
\(R_{h}\) 0.660679 L 0.660727 4.82e-05 0.007% 3.74e-04 5.29e-05 1.06e-04 0.016%
\(V\) 1.00000 L \(^{3}\) 1.00003 2.96e-05 0.003% 1.18e-03 1.67e-04 3.35e-04 0.034%
\(C_{0}\) 0.620351 L 0.620356 5.50e-06 0.001% 2.44e-04 3.45e-05 6.93e-05 0.011%
Eigenvalue of \(\mathbf{S}\) 0.083332 L \(^{2}\) 0.083085 2.47e-04 0.296% 6.53e-05 9.24e-06 1.86e-05 0.022%
Eigenvalue of \(\mathbf{S}\) 0.083333 L \(^{2}\) 0.083329 3.57e-06 0.004% 7.25e-05 1.02e-05 2.06e-05 0.025%
Eigenvalue of \(\mathbf{S}\) 0.083334 L \(^{2}\) 0.083626 2.92e-04 0.350% 7.28e-05 1.03e-05 2.07e-05 0.025%
[\(\sigma\)] 3.6443 1 3.6462 1.89e-03 0.052% 6.87e-03 9.71e-04 1.95e-03 0.054%
[\(\eta\)] 3.0369 1 3.0385 1.59e-03 0.052% 5.72e-03 8.09e-04 1.63e-03 0.054%

Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 0.660679 L 0.660690 1.10e-05 0.002% 1.40e-04 1.98e-05 3.99e-05 0.006%
Eigenvalue of \(\mathbf{\alpha}\) 3.6443 L \(^{3}\) 3.6397 4.56e-03 0.125% 2.53e-03 3.58e-04 7.20e-04 0.020%
Eigenvalue of \(\mathbf{\alpha}\) 3.6443 L \(^{3}\) 3.6441 1.57e-04 0.004% 1.92e-03 2.72e-04 5.47e-04 0.015%
Eigenvalue of \(\mathbf{\alpha}\) 3.6444 L \(^{3}\) 3.6490 4.57e-03 0.126% 2.15e-03 3.04e-04 6.11e-04 0.017%
\(\langle\mathbf{\alpha}\rangle\) 3.64431 L \(^{3}\) 3.64429 2.44e-05 0.001% 1.55e-03 2.20e-04 4.41e-04 0.012%
\(R_{h}\) 0.660679 L 0.660690 1.10e-05 0.002% 1.40e-04 1.98e-05 3.99e-05 0.006%
\(V\) 1.00000 L \(^{3}\) 0.99998 1.91e-05 0.002% 2.85e-04 4.02e-05 8.09e-05 0.008%
\(C_{0}\) 0.620351 L 0.620347 4.45e-06 0.001% 5.88e-05 8.32e-06 1.67e-05 0.003%
Eigenvalue of \(\mathbf{S}\) 0.083332 L \(^{2}\) 0.083266 6.57e-05 0.079% 3.83e-05 5.42e-06 1.09e-05 0.013%
Eigenvalue of \(\mathbf{S}\) 0.083333 L \(^{2}\) 0.083343 9.96e-06 0.012% 3.06e-05 4.32e-06 8.69e-06 0.010%
Eigenvalue of \(\mathbf{S}\) 0.083334 L \(^{2}\) 0.083407 7.34e-05 0.088% 3.28e-05 4.64e-06 9.33e-06 0.011%
[\(\sigma\)] 3.6443 1 3.6444 5.52e-05 0.002% 1.69e-03 2.39e-04 4.80e-04 0.013%
[\(\eta\)] 3.0369 1 3.0370 6.28e-05 0.002% 1.41e-03 1.99e-04 4.00e-04 0.013%

Cuboid of 1 x 2 x 3

Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 1.275784 L 1.275790 5.66e-06 0.000% 8.48e-04 1.20e-04 2.41e-04 0.019%
Eigenvalue of \(\mathbf{\alpha}\) 12.7864 L \(^{3}\) 12.7838 2.63e-03 0.021% 4.75e-02 6.71e-03 1.35e-02 0.106%
Eigenvalue of \(\mathbf{\alpha}\) 26.223 L \(^{3}\) 26.225 1.74e-03 0.007% 7.20e-02 1.02e-02 2.05e-02 0.078%
Eigenvalue of \(\mathbf{\alpha}\) 42.6773 L \(^{3}\) 42.6591 1.82e-02 0.043% 1.17e-01 1.65e-02 3.32e-02 0.078%
\(\langle\mathbf{\alpha}\rangle\) 27.2291 L \(^{3}\) 27.2225 6.57e-03 0.024% 5.31e-02 7.50e-03 1.51e-02 0.055%
\(R_{h}\) 1.275784 L 1.275790 5.66e-06 0.000% 8.48e-04 1.20e-04 2.41e-04 0.019%
\(V\) 6.0000 L \(^{3}\) 6.0003 3.26e-04 0.005% 3.85e-03 5.44e-04 1.09e-03 0.018%
\(C_{0}\) 1.12725 L 1.12727 2.20e-05 0.002% 2.41e-04 3.41e-05 6.85e-05 0.006%
Eigenvalue of \(\mathbf{S}\) 0.083333 L \(^{2}\) 0.083273 5.97e-05 0.072% 1.04e-04 1.47e-05 2.95e-05 0.035%
Eigenvalue of \(\mathbf{S}\) 0.333334 L \(^{2}\) 0.333234 1.00e-04 0.030% 3.06e-04 4.33e-05 8.69e-05 0.026%
Eigenvalue of \(\mathbf{S}\) 0.75000 L \(^{2}\) 0.75007 7.19e-05 0.010% 8.35e-04 1.18e-04 2.37e-04 0.032%
[\(\sigma\)] 4.5382 1 4.5368 1.36e-03 0.030% 9.50e-03 1.34e-03 2.70e-03 0.060%
[\(\eta\)] 3.7334 1 3.7323 1.05e-03 0.028% 7.72e-03 1.09e-03 2.19e-03 0.059%

Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 1.275784 L 1.275789 4.60e-06 0.000% 2.15e-04 3.04e-05 6.11e-05 0.005%
Eigenvalue of \(\mathbf{\alpha}\) 12.7864 L \(^{3}\) 12.7830 3.36e-03 0.026% 1.04e-02 1.47e-03 2.96e-03 0.023%
Eigenvalue of \(\mathbf{\alpha}\) 26.223 L \(^{3}\) 26.219 3.93e-03 0.015% 2.43e-02 3.43e-03 6.90e-03 0.026%
Eigenvalue of \(\mathbf{\alpha}\) 42.6773 L \(^{3}\) 42.6797 2.44e-03 0.006% 4.07e-02 5.76e-03 1.16e-02 0.027%
\(\langle\mathbf{\alpha}\rangle\) 27.2291 L \(^{3}\) 27.2273 1.82e-03 0.007% 1.48e-02 2.09e-03 4.20e-03 0.015%
\(R_{h}\) 1.275784 L 1.275789 4.60e-06 0.000% 2.15e-04 3.04e-05 6.11e-05 0.005%
\(V\) 6.0000 L \(^{3}\) 5.9998 2.09e-04 0.003% 3.64e-03 5.15e-04 1.04e-03 0.017%
\(C_{0}\) 1.12725 L 1.12724 1.14e-05 0.001% 2.28e-04 3.23e-05 6.49e-05 0.006%
Eigenvalue of \(\mathbf{S}\) 0.083333 L \(^{2}\) 0.083337 4.05e-06 0.005% 4.86e-05 6.87e-06 1.38e-05 0.017%
Eigenvalue of \(\mathbf{S}\) 0.333334 L \(^{2}\) 0.333397 6.34e-05 0.019% 1.96e-04 2.78e-05 5.58e-05 0.017%
Eigenvalue of \(\mathbf{S}\) 0.75000 L \(^{2}\) 0.75002 2.27e-05 0.003% 3.64e-04 5.15e-05 1.04e-04 0.014%
[\(\sigma\)] 4.5382 1 4.5380 1.59e-04 0.004% 4.07e-03 5.76e-04 1.16e-03 0.026%
[\(\eta\)] 3.7334 1 3.7333 1.22e-04 0.003% 3.31e-03 4.68e-04 9.41e-04 0.025%

Polymer

Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 2.15962 L 2.15940 2.18e-04 0.010% 2.03e-03 2.87e-04 5.77e-04 0.027%
Eigenvalue of \(\mathbf{\alpha}\) 65.418 L \(^{3}\) 65.457 3.95e-02 0.060% 2.75e-01 3.89e-02 7.83e-02 0.120%
Eigenvalue of \(\mathbf{\alpha}\) 84.087 L \(^{3}\) 84.199 1.12e-01 0.134% 4.53e-01 6.40e-02 1.29e-01 0.153%
Eigenvalue of \(\mathbf{\alpha}\) 270.51 L \(^{3}\) 270.60 8.78e-02 0.032% 9.53e-01 1.35e-01 2.71e-01 0.100%
\(\langle\mathbf{\alpha}\rangle\) 140.004 L \(^{3}\) 140.085 8.09e-02 0.058% 3.55e-01 5.02e-02 1.01e-01 0.072%
\(R_{h}\) 2.15962 L 2.15940 2.18e-04 0.010% 2.03e-03 2.87e-04 5.77e-04 0.027%
\(V\) 17.1020 L \(^{3}\) 17.0846 1.74e-02 0.102% 2.94e-02 4.15e-03 8.34e-03 0.049%
\(C_{0}\) 1.598280 L 1.597738 5.42e-04 0.034% 9.15e-04 1.29e-04 2.60e-04 0.016%
Eigenvalue of \(\mathbf{S}\) 0.415410 L \(^{2}\) 0.415514 1.04e-04 0.025% 8.09e-04 1.14e-04 2.30e-04 0.055%
Eigenvalue of \(\mathbf{S}\) 0.64270 L \(^{2}\) 0.64369 9.92e-04 0.154% 2.39e-03 3.38e-04 6.79e-04 0.106%
Eigenvalue of \(\mathbf{S}\) 3.6002 L \(^{2}\) 3.5990 1.18e-03 0.033% 6.15e-03 8.69e-04 1.75e-03 0.049%
[\(\sigma\)] 8.1864 1 8.1995 1.31e-02 0.160% 2.41e-02 3.40e-03 6.84e-03 0.084%
[\(\eta\)] 6.6823 1 6.6932 1.09e-02 0.163% 1.92e-02 2.72e-03 5.46e-03 0.082%

Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 2.15962 L 2.15971 8.59e-05 0.004% 6.88e-04 9.73e-05 1.96e-04 0.009%
Eigenvalue of \(\mathbf{\alpha}\) 65.418 L \(^{3}\) 65.439 2.07e-02 0.032% 9.77e-02 1.38e-02 2.78e-02 0.042%
Eigenvalue of \(\mathbf{\alpha}\) 84.087 L \(^{3}\) 84.075 1.18e-02 0.014% 1.31e-01 1.85e-02 3.72e-02 0.044%
Eigenvalue of \(\mathbf{\alpha}\) 270.51 L \(^{3}\) 270.53 1.92e-02 0.007% 3.30e-01 4.66e-02 9.37e-02 0.035%
\(\langle\mathbf{\alpha}\rangle\) 140.004 L \(^{3}\) 140.014 1.03e-02 0.007% 1.28e-01 1.81e-02 3.64e-02 0.026%
\(R_{h}\) 2.15962 L 2.15971 8.59e-05 0.004% 6.88e-04 9.73e-05 1.96e-04 0.009%
\(V\) 17.1020 L \(^{3}\) 17.1020 2.36e-05 0.000% 2.01e-02 2.84e-03 5.70e-03 0.033%
\(C_{0}\) 1.598280 L 1.598280 6.00e-08 0.000% 6.25e-04 8.84e-05 1.78e-04 0.011%
Eigenvalue of \(\mathbf{S}\) 0.415410 L \(^{2}\) 0.415199 2.11e-04 0.051% 4.34e-04 6.13e-05 1.23e-04 0.030%
Eigenvalue of \(\mathbf{S}\) 0.64270 L \(^{2}\) 0.64290 2.02e-04 0.031% 9.08e-04 1.28e-04 2.58e-04 0.040%
Eigenvalue of \(\mathbf{S}\) 3.6002 L \(^{2}\) 3.6026 2.37e-03 0.066% 4.45e-03 6.29e-04 1.26e-03 0.035%
[\(\sigma\)] 8.1864 1 8.1870 6.15e-04 0.008% 1.20e-02 1.69e-03 3.40e-03 0.042%
[\(\eta\)] 6.6823 1 6.6828 5.38e-04 0.008% 9.67e-03 1.37e-03 2.75e-03 0.041%

Protein

Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 21.4869 A 21.4850 1.85e-03 0.009% 1.29e-02 1.83e-03 3.68e-03 0.017%
Eigenvalue of \(\mathbf{\alpha}\) 96681 A \(^{3}\) 96662 2e+01 0.019% 3e+02 4e+01 8e+01 0.082%
Eigenvalue of \(\mathbf{\alpha}\) 100999 A \(^{3}\) 101047 5e+01 0.047% 3e+02 4e+01 8e+01 0.084%
Eigenvalue of \(\mathbf{\alpha}\) 184488 A \(^{3}\) 184499 1e+01 0.006% 5e+02 8e+01 2e+02 0.082%
\(\langle\mathbf{\alpha}\rangle\) 127389 A \(^{3}\) 127403 1e+01 0.011% 2e+02 3e+01 6e+01 0.046%
\(R_{h}\) 21.4869 A 21.4850 1.85e-03 0.009% 1.29e-02 1.83e-03 3.68e-03 0.017%
\(V\) 32213.6 A \(^{3}\) 32283.2 7e+01 0.216% 2e+01 3e+00 7e+00 0.022%
\(C_{0}\) 19.7386 A 19.7528 1.42e-02 0.072% 4.99e-03 7.06e-04 1.42e-03 0.007%
Eigenvalue of \(\mathbf{S}\) 58.9754 A \(^{2}\) 58.9957 2.03e-02 0.034% 1.61e-01 2.27e-02 4.57e-02 0.078%
Eigenvalue of \(\mathbf{S}\) 62.500 A \(^{2}\) 62.611 1.11e-01 0.178% 5.96e-02 8.43e-03 1.69e-02 0.027%
Eigenvalue of \(\mathbf{S}\) 184.325 A \(^{2}\) 184.366 4.05e-02 0.022% 3.16e-01 4.47e-02 8.98e-02 0.049%
[\(\sigma\)] 3.9545 1 3.9464 8.09e-03 0.205% 7.03e-03 9.94e-04 2.00e-03 0.051%
[\(\eta\)] 3.2907 1 3.2839 6.76e-03 0.206% 5.84e-03 8.26e-04 1.66e-03 0.050%

Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 21.4869 A 21.4857 1.23e-03 0.006% 3.67e-03 5.19e-04 1.04e-03 0.005%
Eigenvalue of \(\mathbf{\alpha}\) 96681 A \(^{3}\) 96673 8e+00 0.008% 7e+01 1e+01 2e+01 0.022%
Eigenvalue of \(\mathbf{\alpha}\) 100999 A \(^{3}\) 100989 1e+01 0.010% 9e+01 1e+01 2e+01 0.024%
Eigenvalue of \(\mathbf{\alpha}\) 184488 A \(^{3}\) 184490 2e+00 0.001% 1e+02 2e+01 4e+01 0.021%
\(\langle\mathbf{\alpha}\rangle\) 127389 A \(^{3}\) 127384 5e+00 0.004% 5e+01 7e+00 1e+01 0.011%
\(R_{h}\) 21.4869 A 21.4857 1.23e-03 0.006% 3.67e-03 5.19e-04 1.04e-03 0.005%
\(V\) 32213.6 A \(^{3}\) 32216.1 2e+00 0.008% 1e+01 2e+00 4e+00 0.012%
\(C_{0}\) 19.7386 A 19.7391 5.32e-04 0.003% 2.75e-03 3.89e-04 7.83e-04 0.004%
Eigenvalue of \(\mathbf{S}\) 58.9754 A \(^{2}\) 58.9851 9.66e-03 0.016% 3.56e-02 5.04e-03 1.01e-02 0.017%
Eigenvalue of \(\mathbf{S}\) 62.500 A \(^{2}\) 62.519 1.86e-02 0.030% 4.16e-02 5.89e-03 1.18e-02 0.019%
Eigenvalue of \(\mathbf{S}\) 184.325 A \(^{2}\) 184.320 5.34e-03 0.003% 8.61e-02 1.22e-02 2.45e-02 0.013%
[\(\sigma\)] 3.9545 1 3.9541 4.41e-04 0.011% 2.02e-03 2.86e-04 5.74e-04 0.015%
[\(\eta\)] 3.2907 1 3.2903 4.12e-04 0.013% 1.68e-03 2.38e-04 4.77e-04 0.015%

Torus with a minor radius of 1 and major radius of 4 sphere representation

Comparison with same representation; Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 3.72748 L 3.72734 1.44e-04 0.004% 1.76e-03 2.49e-04 5.01e-04 0.013%
Eigenvalue of \(\mathbf{\alpha}\) 156.488 L \(^{3}\) 156.565 7.70e-02 0.049% 5.27e-01 7.45e-02 1.50e-01 0.096%
Eigenvalue of \(\mathbf{\alpha}\) 972.069 L \(^{3}\) 969.914 2.16e+00 0.222% 1.99e+00 2.82e-01 5.66e-01 0.058%
Eigenvalue of \(\mathbf{\alpha}\) 972.08 L \(^{3}\) 974.21 2.1e+00 0.219% 2.4e+00 3.4e-01 6.8e-01 0.070%
\(\langle\mathbf{\alpha}\rangle\) 700.212 L \(^{3}\) 700.229 1.65e-02 0.002% 1.29e+00 1.83e-01 3.68e-01 0.053%
\(R_{h}\) 3.72748 L 3.72734 1.44e-04 0.004% 1.76e-03 2.49e-04 5.01e-04 0.013%
\(V\) 78.932 L \(^{3}\) 78.917 1.51e-02 0.019% 1.34e-01 1.90e-02 3.82e-02 0.048%
\(C_{0}\) 2.66106 L 2.66089 1.69e-04 0.006% 1.51e-03 2.14e-04 4.29e-04 0.016%
Eigenvalue of \(\mathbf{S}\) 0.249925 L \(^{2}\) 0.249769 1.56e-04 0.063% 5.82e-04 8.23e-05 1.65e-04 0.066%
Eigenvalue of \(\mathbf{S}\) 8.37472 L \(^{2}\) 8.36287 1.19e-02 0.142% 7.57e-03 1.07e-03 2.15e-03 0.026%
Eigenvalue of \(\mathbf{S}\) 8.3748 L \(^{2}\) 8.3838 8.97e-03 0.107% 5.71e-03 8.07e-04 1.62e-03 0.019%
[\(\sigma\)] 8.8711 1 8.8730 1.91e-03 0.022% 2.30e-02 3.26e-03 6.55e-03 0.074%
[\(\eta\)] 7.2131 1 7.2148 1.73e-03 0.024% 1.86e-02 2.63e-03 5.29e-03 0.073%

Comparison with same representation; Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 3.72748 L 3.72740 7.68e-05 0.002% 8.37e-04 1.18e-04 2.38e-04 0.006%
Eigenvalue of \(\mathbf{\alpha}\) 156.488 L \(^{3}\) 156.443 4.52e-02 0.029% 2.02e-01 2.85e-02 5.73e-02 0.037%
Eigenvalue of \(\mathbf{\alpha}\) 972.069 L \(^{3}\) 971.379 6.90e-01 0.071% 6.28e-01 8.88e-02 1.78e-01 0.018%
Eigenvalue of \(\mathbf{\alpha}\) 972.08 L \(^{3}\) 972.79 7.12e-01 0.073% 7.80e-01 1.10e-01 2.22e-01 0.023%
\(\langle\mathbf{\alpha}\rangle\) 700.212 L \(^{3}\) 700.205 7.36e-03 0.001% 4.19e-01 5.92e-02 1.19e-01 0.017%
\(R_{h}\) 3.72748 L 3.72740 7.68e-05 0.002% 8.37e-04 1.18e-04 2.38e-04 0.006%
\(V\) 78.932 L \(^{3}\) 78.958 2.62e-02 0.033% 4.98e-02 7.05e-03 1.42e-02 0.018%
\(C_{0}\) 2.66106 L 2.66136 2.96e-04 0.011% 5.60e-04 7.92e-05 1.59e-04 0.006%
Eigenvalue of \(\mathbf{S}\) 0.249925 L \(^{2}\) 0.249834 9.05e-05 0.036% 1.72e-04 2.43e-05 4.88e-05 0.020%
Eigenvalue of \(\mathbf{S}\) 8.37472 L \(^{2}\) 8.36806 6.66e-03 0.080% 3.64e-03 5.14e-04 1.03e-03 0.012%
Eigenvalue of \(\mathbf{S}\) 8.3748 L \(^{2}\) 8.3815 6.69e-03 0.080% 3.37e-03 4.77e-04 9.59e-04 0.011%
[\(\sigma\)] 8.8711 1 8.8680 3.06e-03 0.034% 8.46e-03 1.20e-03 2.41e-03 0.027%
[\(\eta\)] 7.2131 1 7.2106 2.45e-03 0.034% 6.88e-03 9.73e-04 1.96e-03 0.027%

Comparison to analytic values such that representation generates error; Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 3.72768 L 3.72734 3.44e-04 0.009% 1.76e-03 2.49e-04 5.01e-04 0.013%
Eigenvalue of \(\mathbf{\alpha}\) 156.53 L \(^{3}\) 156.56 3.50e-02 0.022% 5.27e-01 7.45e-02 1.50e-01 0.096%
Eigenvalue of \(\mathbf{\alpha}\) 972.21 L \(^{3}\) 969.91 2.3e+00 0.236% 2.0e+00 2.8e-01 5.7e-01 0.058%
Eigenvalue of \(\mathbf{\alpha}\) 972.21 L \(^{3}\) 974.21 2.0e+00 0.205% 2.4e+00 3.4e-01 6.8e-01 0.070%
\(\langle\mathbf{\alpha}\rangle\) 700.31 L \(^{3}\) 700.23 8.15e-02 0.012% 1.29e+00 1.83e-01 3.68e-01 0.053%
\(R_{h}\) 3.72768 L 3.72734 3.44e-04 0.009% 1.76e-03 2.49e-04 5.01e-04 0.013%
\(V\) 78.956835 L \(^{3}\) 78.916912 3.99e-02 0.051% 1.34e-01 1.90e-02 3.82e-02 0.048%
\(C_{0}\) 2.66134 L 2.66089 4.49e-04 0.017% 1.51e-03 2.14e-04 4.29e-04 0.016%
Eigenvalue of \(\mathbf{S}\) 0.250000 L \(^{2}\) 0.249769 2.31e-04 0.093% 5.82e-04 8.23e-05 1.65e-04 0.066%
Eigenvalue of \(\mathbf{S}\) 8.375000 L \(^{2}\) 8.362865 1.21e-02 0.145% 7.57e-03 1.07e-03 2.15e-03 0.026%
Eigenvalue of \(\mathbf{S}\) 8.375000 L \(^{2}\) 8.383773 8.77e-03 0.105% 5.71e-03 8.07e-04 1.62e-03 0.019%
[\(\sigma\)] 8.8696 1 8.8730 3.41e-03 0.038% 2.30e-02 3.26e-03 6.55e-03 0.074%

Comparison to analytic values such that representation generates error; Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 3.72768 L 3.72740 2.77e-04 0.007% 8.37e-04 1.18e-04 2.38e-04 0.006%
Eigenvalue of \(\mathbf{\alpha}\) 156.53 L \(^{3}\) 156.44 8.72e-02 0.056% 2.02e-01 2.85e-02 5.73e-02 0.037%
Eigenvalue of \(\mathbf{\alpha}\) 972.21 L \(^{3}\) 971.38 8.31e-01 0.085% 6.28e-01 8.88e-02 1.78e-01 0.018%
Eigenvalue of \(\mathbf{\alpha}\) 972.21 L \(^{3}\) 972.79 5.82e-01 0.060% 7.80e-01 1.10e-01 2.22e-01 0.023%
\(\langle\mathbf{\alpha}\rangle\) 700.31 L \(^{3}\) 700.20 1.05e-01 0.015% 4.19e-01 5.92e-02 1.19e-01 0.017%
\(R_{h}\) 3.72768 L 3.72740 2.77e-04 0.007% 8.37e-04 1.18e-04 2.38e-04 0.006%
\(V\) 78.956835 L \(^{3}\) 78.958221 1.39e-03 0.002% 4.98e-02 7.05e-03 1.42e-02 0.018%
\(C_{0}\) 2.66134 L 2.66136 1.56e-05 0.001% 5.60e-04 7.92e-05 1.59e-04 0.006%
Eigenvalue of \(\mathbf{S}\) 0.250000 L \(^{2}\) 0.249834 1.66e-04 0.066% 1.72e-04 2.43e-05 4.88e-05 0.020%
Eigenvalue of \(\mathbf{S}\) 8.375000 L \(^{2}\) 8.368057 6.94e-03 0.083% 3.64e-03 5.14e-04 1.03e-03 0.012%
Eigenvalue of \(\mathbf{S}\) 8.375000 L \(^{2}\) 8.381487 6.49e-03 0.077% 3.37e-03 4.77e-04 9.59e-04 0.011%
[\(\sigma\)] 8.8696 1 8.8680 1.56e-03 0.018% 8.46e-03 1.20e-03 2.41e-03 0.027%

Two touching spheres one of radius 1 and one of radius 4 voxelized

Comparison with same representation; Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 4.08690 L 4.08690 1.94e-06 0.000% 1.92e-03 2.72e-04 5.46e-04 0.013%
Eigenvalue of \(\mathbf{\alpha}\) 811.97 L \(^{3}\) 810.13 1.8e+00 0.227% 1.7e+00 2.4e-01 4.8e-01 0.059%
Eigenvalue of \(\mathbf{\alpha}\) 811.99 L \(^{3}\) 813.97 2.0e+00 0.244% 1.8e+00 2.5e-01 5.0e-01 0.062%
Eigenvalue of \(\mathbf{\alpha}\) 957.90 L \(^{3}\) 958.31 4.14e-01 0.043% 1.81e+00 2.56e-01 5.15e-01 0.054%
\(\langle\mathbf{\alpha}\rangle\) 860.62 L \(^{3}\) 860.80 1.82e-01 0.021% 9.73e-01 1.38e-01 2.77e-01 0.032%
\(R_{h}\) 4.08690 L 4.08690 1.94e-06 0.000% 1.92e-03 2.72e-04 5.46e-04 0.013%
\(V\) 272.254 L \(^{3}\) 272.191 6.27e-02 0.023% 1.30e-01 1.84e-02 3.70e-02 0.014%
\(C_{0}\) 4.02064 L 4.02033 3.08e-04 0.008% 6.42e-04 9.07e-05 1.82e-04 0.005%
Eigenvalue of \(\mathbf{S}\) 3.15370 L \(^{2}\) 3.14655 7.15e-03 0.227% 1.65e-03 2.34e-04 4.70e-04 0.015%
Eigenvalue of \(\mathbf{S}\) 3.15374 L \(^{2}\) 3.15678 3.04e-03 0.096% 5.15e-03 7.28e-04 1.46e-03 0.046%
Eigenvalue of \(\mathbf{S}\) 3.53277 L \(^{2}\) 3.53288 1.10e-04 0.003% 4.10e-03 5.80e-04 1.17e-03 0.033%
[\(\sigma\)] 3.1611 1 3.1625 1.39e-03 0.044% 3.65e-03 5.17e-04 1.04e-03 0.033%
[\(\eta\)] 2.6328 1 2.6339 1.12e-03 0.043% 3.05e-03 4.32e-04 8.68e-04 0.033%

Comparison with same representation; Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 4.08690 L 4.08673 1.70e-04 0.004% 5.93e-04 8.39e-05 1.69e-04 0.004%
Eigenvalue of \(\mathbf{\alpha}\) 811.97 L \(^{3}\) 811.42 5.46e-01 0.067% 4.68e-01 6.62e-02 1.33e-01 0.016%
Eigenvalue of \(\mathbf{\alpha}\) 811.99 L \(^{3}\) 812.46 4.73e-01 0.058% 4.94e-01 6.99e-02 1.40e-01 0.017%
Eigenvalue of \(\mathbf{\alpha}\) 957.90 L \(^{3}\) 957.92 2.48e-02 0.003% 6.20e-01 8.76e-02 1.76e-01 0.018%
\(\langle\mathbf{\alpha}\rangle\) 860.62 L \(^{3}\) 860.60 1.60e-02 0.002% 3.44e-01 4.87e-02 9.78e-02 0.011%
\(R_{h}\) 4.08690 L 4.08673 1.70e-04 0.004% 5.93e-04 8.39e-05 1.69e-04 0.004%
\(V\) 272.254 L \(^{3}\) 272.285 3.06e-02 0.011% 7.27e-02 1.03e-02 2.07e-02 0.008%
\(C_{0}\) 4.02064 L 4.02079 1.51e-04 0.004% 3.58e-04 5.06e-05 1.02e-04 0.003%
Eigenvalue of \(\mathbf{S}\) 3.15370 L \(^{2}\) 3.15247 1.23e-03 0.039% 1.09e-03 1.55e-04 3.11e-04 0.010%
Eigenvalue of \(\mathbf{S}\) 3.15374 L \(^{2}\) 3.15508 1.34e-03 0.042% 1.20e-03 1.70e-04 3.41e-04 0.011%
Eigenvalue of \(\mathbf{S}\) 3.53277 L \(^{2}\) 3.53334 5.68e-04 0.016% 1.91e-03 2.70e-04 5.43e-04 0.015%
[\(\sigma\)] 3.1611 1 3.1607 4.21e-04 0.013% 1.36e-03 1.92e-04 3.86e-04 0.012%
[\(\eta\)] 2.6328 1 2.6324 3.94e-04 0.015% 1.13e-03 1.60e-04 3.22e-04 0.012%

Comparison to analytic values such that representation generates error; Number of walks = 1e6; Number of interior samples = 1e6

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 4.07968 L 4.08690 7.22e-03 0.177% 1.92e-03 2.72e-04 5.46e-04 0.013%
Eigenvalue of \(\mathbf{\alpha}\) 807.744 L \(^{3}\) 810.125 2.38e+00 0.295% 1.68e+00 2.38e-01 4.78e-01 0.059%
Eigenvalue of \(\mathbf{\alpha}\) 807.744 L \(^{3}\) 813.968 6.22e+00 0.771% 1.76e+00 2.49e-01 5.00e-01 0.062%
Eigenvalue of \(\mathbf{\alpha}\) 952.64 L \(^{3}\) 958.31 5.7e+00 0.596% 1.8e+00 2.6e-01 5.1e-01 0.054%
\(\langle\mathbf{\alpha}\rangle\) 856.064 L \(^{3}\) 860.802 4.74e+00 0.553% 9.73e-01 1.38e-01 2.77e-01 0.032%
\(R_{h}\) 4.07968 L 4.08690 7.22e-03 0.177% 1.92e-03 2.72e-04 5.46e-04 0.013%
\(V\) 272.27136 L \(^{3}\) 272.19131 8.00e-02 0.029% 1.30e-01 1.84e-02 3.70e-02 0.014%
\(C_{0}\) 4.02072 L 4.02033 3.88e-04 0.010% 6.42e-04 9.07e-05 1.82e-04 0.005%
Eigenvalue of \(\mathbf{S}\) 3.15384 L \(^{2}\) 3.14655 7.29e-03 0.231% 1.65e-03 2.34e-04 4.70e-04 0.015%
Eigenvalue of \(\mathbf{S}\) 3.15384 L \(^{2}\) 3.15678 2.94e-03 0.093% 5.15e-03 7.28e-04 1.46e-03 0.046%
Eigenvalue of \(\mathbf{S}\) 3.532544 L \(^{2}\) 3.532880 3.36e-04 0.009% 4.10e-03 5.80e-04 1.17e-03 0.033%
[\(\sigma\)] 3.1442 1 3.1625 1.83e-02 0.582% 3.65e-03 5.17e-04 1.04e-03 0.033%

Comparison to analytic values such that representation generates error; Number of walks = 1e7; Number of interior samples = 1e7

Property Ground Truth Units Mean Difference Rel. Diff. Std. Dev. Std. Uncert. Expand Uncert. Rel. Exp. Uncert.
\(C\) 4.07968 L 4.08673 7.05e-03 0.173% 5.93e-04 8.39e-05 1.69e-04 0.004%
Eigenvalue of \(\mathbf{\alpha}\) 807.744 L \(^{3}\) 811.424 3.68e+00 0.456% 4.68e-01 6.62e-02 1.33e-01 0.016%
Eigenvalue of \(\mathbf{\alpha}\) 807.744 L \(^{3}\) 812.463 4.72e+00 0.584% 4.94e-01 6.99e-02 1.40e-01 0.017%
Eigenvalue of \(\mathbf{\alpha}\) 952.64 L \(^{3}\) 957.92 5.3e+00 0.555% 6.2e-01 8.8e-02 1.8e-01 0.018%
\(\langle\mathbf{\alpha}\rangle\) 856.064 L \(^{3}\) 860.604 4.54e+00 0.530% 3.44e-01 4.87e-02 9.78e-02 0.011%
\(R_{h}\) 4.07968 L 4.08673 7.05e-03 0.173% 5.93e-04 8.39e-05 1.69e-04 0.004%
\(V\) 272.27136 L \(^{3}\) 272.28459 1.32e-02 0.005% 7.27e-02 1.03e-02 2.07e-02 0.008%
\(C_{0}\) 4.02072 L 4.02079 7.08e-05 0.002% 3.58e-04 5.06e-05 1.02e-04 0.003%
Eigenvalue of \(\mathbf{S}\) 3.15384 L \(^{2}\) 3.15247 1.37e-03 0.044% 1.09e-03 1.55e-04 3.11e-04 0.010%
Eigenvalue of \(\mathbf{S}\) 3.15384 L \(^{2}\) 3.15508 1.24e-03 0.039% 1.20e-03 1.70e-04 3.41e-04 0.011%
Eigenvalue of \(\mathbf{S}\) 3.532544 L \(^{2}\) 3.533338 7.94e-04 0.022% 1.91e-03 2.70e-04 5.43e-04 0.015%
[\(\sigma\)] 3.1442 1 3.1607 1.65e-02 0.524% 1.36e-03 1.92e-04 3.86e-04 0.012%